Average Error: 3.6 → 1.4
Time: 3.7m
Precision: 64
Internal Precision: 1088
\[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c_n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{1 + e^{-s}} \le 0.49183472926145494 \lor \neg \left(\frac{1}{1 + e^{-s}} \le 0.5000000000000013\right):\\ \;\;\;\;\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p}}{c_p \cdot \left(t \cdot \frac{1}{2} + \log \frac{1}{2}\right) + 1} \cdot \frac{{\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}{{\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c_n}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(c_p - c_n\right) \cdot \left(\frac{1}{2} \cdot s\right)\\ \end{array}\]

Error

Bits error versus c_p

Bits error versus c_n

Bits error versus t

Bits error versus s

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original3.6
Target1.9
Herbie1.4
\[{\left(\frac{1 + e^{-t}}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(\frac{1 + e^{t}}{1 + e^{s}}\right)}^{c_n}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ 1 (+ 1 (exp (- s)))) < 0.49183472926145494 or 0.5000000000000013 < (/ 1 (+ 1 (exp (- s))))

    1. Initial program 3.9

      \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c_n}}\]

    2. Taylor expanded around 0 1.9

      \[\leadsto \frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}{\color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(t \cdot c_p\right) + \log \frac{1}{2} \cdot c_p\right)\right)} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c_n}}\]

    3. Applied simplify1.9

      \[\leadsto \color{blue}{\frac{{\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}{{\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c_n}} \cdot \frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p}}{c_p \cdot \left(\log \frac{1}{2} + t \cdot \frac{1}{2}\right) + 1}}\]

    if 0.49183472926145494 < (/ 1 (+ 1 (exp (- s)))) < 0.5000000000000013

    1. Initial program 3.3

      \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c_n}}\]

    2. Taylor expanded around 0 0.8

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(s \cdot c_p\right) + 1\right) - \frac{1}{2} \cdot \left(c_n \cdot s\right)}\]

    3. Applied simplify0.8

      \[\leadsto \color{blue}{1 + \left(c_p - c_n\right) \cdot \left(s \cdot \frac{1}{2}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Applied simplify1.4

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{1}{1 + e^{-s}} \le 0.49183472926145494 \lor \neg \left(\frac{1}{1 + e^{-s}} \le 0.5000000000000013\right):\\ \;\;\;\;\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p}}{c_p \cdot \left(t \cdot \frac{1}{2} + \log \frac{1}{2}\right) + 1} \cdot \frac{{\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}{{\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c_n}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(c_p - c_n\right) \cdot \left(\frac{1}{2} \cdot s\right)\\ \end{array}}\]

Runtime

Time bar (total: 3.7m)Debug logProfile

herbie shell --seed 2018167 
(FPCore (c_p c_n t s)
  :name "Harley's example"
  :pre (and (< 0 c_p) (< 0 c_n))

  :herbie-target
  (* (pow (/ (+ 1 (exp (- t))) (+ 1 (exp (- s)))) c_p) (pow (/ (+ 1 (exp t)) (+ 1 (exp s))) c_n))

  (/ (* (pow (/ 1 (+ 1 (exp (- s)))) c_p) (pow (- 1 (/ 1 (+ 1 (exp (- s))))) c_n)) (* (pow (/ 1 (+ 1 (exp (- t)))) c_p) (pow (- 1 (/ 1 (+ 1 (exp (- t))))) c_n))))